This work is organized as follows. In § we recall some general facts about
moduli spaces of sheaves on \({P}^2\) , the structure of the derived category \(D^b\operatorname{Coh}(K_{{P}^2})\) 
and the space of Bridgeland stability conditions associated to it.
In § we recall the definition and main properties
of the scattering diagram for quivers with potentials, extend this notion to general triangulated categories, and explain the relation between scattering rays and attractor flow trees.
In §, we revisit the scattering diagram \(\mathcal {D}_\psi ^{\rm LV}\) 
for the large volume central charge constructed in [1]}, generalize it to arbitrary
\(\psi \) , give an effective algorithm for determining the possible initial rays contributing to the index,
and illustrate this procedure by computing the Gieseker index for various Chern vectors of rank 0 or 1. In §, we prove the Attractor Conjecture for the orbifold quiver, construct the corresponding scattering diagram \(\mathcal {D}_o\)  associated to the orbifold quiver, and determine the scattering
sequences contributing to the quiver index for various dimension vectors.  In §, we
determine the scattering diagram \(\mathcal {D}^{\Pi }_\psi \)  on the slice of \(\Pi \) -stability conditions, and outline the proof of the SAFC in this case. Details of the proof will appear in [2]}.
In Appendix  we derive the Eichler integral representation (REF )
of the periods \(T,T_D\)  provided by local mirror symmetry, and use it to obtain expansions
around the large volume, conifold and orbifold points. In § we determine
the object in \(D^b \operatorname{Coh}_c K_{{P}^2}\)  which becomes massless at the orbifold point \(\tau =p/q\) 
for low values of \(p,q\) . In §, we provide some details
on the mathematical definition of DT invariants.
In § we give further examples of scattering sequences
for higher rank sheaves at large volume, complementing the examples in §REF .
Finally, in § we describe the main features of a
Mathematica package which we have developed in the course of this investigation, which is
freely available for further explorations.
